Question: Determine how many solutions exist for the system of equations. ${-6x-3y = -24}$ ${2x+y = 8}$
Explanation: Convert both equations to slope-intercept form: ${-6x-3y = -24}$ $-6x{+6x} - 3y = -24{+6x}$ $-3y = -24+6x$ $y = 8-2x$ ${y = -2x+8}$ ${2x+y = 8}$ $2x{-2x} + y = 8{-2x}$ $y = 8-2x$ ${y = -2x+8}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -2x+8}$ ${y = -2x+8}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-6x-3y = -24}$ is also a solution of ${2x+y = 8}$, there are infinitely many solutions.